Sure, let's analyze the function [tex]\( f(x) = \frac{5x - 2}{2x + 1} \)[/tex].
### Step-by-Step Solution:
#### 1. Understanding the function:
The function [tex]\( f(x) = \frac{5x - 2}{2x + 1} \)[/tex] is a rational function, which is defined as the ratio of two polynomials.
#### 2. Identify the components of the function:
- Numerator: [tex]\( 5x - 2 \)[/tex]
- Denominator: [tex]\( 2x + 1 \)[/tex]
#### 3. Domain of the function:
The domain of a rational function is all real numbers except for those where the denominator is zero. To find these values, set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 1 = 0 \][/tex]
[tex]\[ 2x = -1 \][/tex]
[tex]\[ x = -\frac{1}{2} \][/tex]
Thus, the domain of [tex]\( f(x) \)[/tex] is all real numbers except [tex]\( x = -\frac{1}{2} \)[/tex].
#### 4. Vertical asymptote:
A vertical asymptote occurs where the denominator is zero (provided the numerator is not zero at that point). From the previous step, we know the denominator is zero at [tex]\( x = -\frac{1}{2} \)[/tex]. Therefore, there is a vertical asymptote at [tex]\( x = -\frac{1}{2} \)[/tex].
#### 5. Horizontal asymptote:
To find the horizontal asymptote for a rational function, compare the degrees of the numerator and the denominator:
- Degree of numerator ( [tex]\( 5x - 2 \)[/tex] ): 1
- Degree of denominator ( [tex]\( 2x + 1 \)[/tex] ): 1
Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is found by dividing the leading coefficients:
[tex]\[ y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{5}{2} \][/tex]
So, the horizontal asymptote is [tex]\( y = \frac{5}{2} \)[/tex].
#### 6. Intercepts:
- y-intercept: To find the y-intercept, set [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{5(0) - 2}{2(0) + 1} = \frac{-2}{1} = -2 \][/tex]
So, the y-intercept is [tex]\( (0, -2) \)[/tex].
- x-intercept: To find the x-intercept, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \frac{5x - 2}{2x + 1} = 0 \][/tex]
[tex]\[ 5x - 2 = 0 \][/tex]
[tex]\[ 5x = 2 \][/tex]
[tex]\[ x = \frac{2}{5} \][/tex]
So, the x-intercept is [tex]\( \left(\frac{2}{5}, 0\right) \)[/tex].
### Summary:
- Domain: [tex]\( x \in \mathbb{R} \setminus \left\{-\frac{1}{2}\right\} \)[/tex]
- Vertical asymptote: [tex]\( x = -\frac{1}{2} \)[/tex]
- Horizontal asymptote: [tex]\( y = \frac{5}{2} \)[/tex]
- y-intercept: [tex]\( (0, -2) \)[/tex]
- x-intercept: [tex]\( \left(\frac{2}{5}, 0\right) \)[/tex]
This is a complete analysis of the function [tex]\( f(x) = \frac{5x - 2}{2x + 1} \)[/tex].
